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# If a and b are both positive even integers which of the following must

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Math Expert
Joined: 02 Sep 2009
Posts: 51121
If a and b are both positive even integers which of the following must  [#permalink]

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08 Jun 2018, 11:11
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5% (low)

Question Stats:

91% (00:45) correct 9% (01:05) wrong based on 78 sessions

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If $$a$$ and $$b$$ are both positive even integers, which of the following must be even?

I. $$a^b$$

II. $$(a+1)^b$$

III. $$a^{(b+1)}$$

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III

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Posts: 44
Re: If a and b are both positive even integers which of the following must  [#permalink]

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08 Jun 2018, 11:19
Even number to the power Even is always even. Hence I is true.

Odd number to the power Even is Odd. In fact odd number to the power Even/Odd is always odd. Hence II is False

Even number to the power Odd is Even. Hence III is True.

Option D.
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Posts: 2291
Re: If a and b are both positive even integers which of the following must  [#permalink]

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08 Jun 2018, 11:32

Solution

Given:
• a and b are both positive even integers

To find:
• Out of the given options, which of them are always even

Approach and Working:
In this case, we only know both a and b individually are even numbers

Now, considering the expressions one by one,
• $$a^b = (even)^{even} = even$$
• $$(a + 1)^b = (odd)^{even} = odd$$
• $$a^{(b+1)} = (even)^{odd} = even$$

Therefore, 1 and 3 will be always even.

Hence, the correct answer is option D.

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Re: If a and b are both positive even integers which of the following must  [#permalink]

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08 Jun 2018, 17:59
Bunuel wrote:
If $$a$$ and $$b$$ are both positive even integers, which of the following must be even?

I. $$a^b$$

II. $$(a+1)^b$$

III. $$a^{(b+1)}$$

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III

Given a>0, b>0 & a,b are even.

Let's say a=2. b=4

I. $$a^b$$=$$2^4$$=16 which is even.

II. $$(a+1)^b$$=$$3^4$$=81 which is not even.

III. $$a^{(b+1)}$$=$$2^5$$=32 which is even

Hence ans option D.
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If a and b are both positive even integers which of the following must  [#permalink]

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10 Jun 2018, 13:50
Bunuel wrote:
If $$a$$ and $$b$$ are both positive even integers, which of the following must be even?

I. $$a^b$$

II. $$(a+1)^b$$

III. $$a^{(b+1)}$$

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III

It's a fundamental question based on rules of exponents. Why do we use exponent? it's a short cut multiplication.

here are some rules:
1.even integer having even power will always yield even. For example: 2*2*2*2*2*2*2*2............ultimate results will be even.
2. An odd integer having odd or even power will always yield odd integer. 3*3*3*3 = 81 which is an odd.
3. An even integer having odd or even power will always yield even integer. 2*2*2*2*2 = 32 even .

Thus D is the best answer.
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Re: If a and b are both positive even integers which of the following must  [#permalink]

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12 Jun 2018, 03:17
I go with D because
1. even^even is always even, try with numbers 2 and 4 (works)
2. odd^even is odd, try with 3^2 (doesnt work)
3. even^off is even, try with 2 and 3 (works).
Conditions 1 and 3 satisfy the question asked, hence answer would be D
Re: If a and b are both positive even integers which of the following must &nbs [#permalink] 12 Jun 2018, 03:17
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